your model has a few issues, and it's not obvious (to me, but I'm no expert) how to fix them.

I think what may be useful is to suggest how to make the model again (in general terms), so that it does what you want. This is just how I would do it, there are undoubtedly many ways, and probably many better....

1. Use the original cube and make it the size you want;
2. Go into Edit mode and select all Vertices (The 'A' key toggles between all and none selected);
3. From the Mesh Tolls menu use Subdivide several times. This will split the surface of the cube into many squares;
4. Deselect all faces/vertices ('A');
5. For each of the face features select the appropriate model faces. E.g. for the nose select the ones in the middle of the cube's face;
6. Extrude those faces as you wish;
7. Do the same for the ears, but extruding twice;
8. Go back to Object Mode;
9. Add a correctly sized UV Sphere and position it;
10. Select the modified Cube again, and add a Boolean Modifier: you want the Union operator and the UV SPhere Object;
11. Apply the Modifier;
12. Repeat for the other side;

What you now have is a 3d model that should be uploadable to Shapeways. Select just the Cube object and try! (You can delete or ignore the spheres now.)

But you hollowed your model which is good practice. An easy way to do this is as follows:
a. Go into Edit Mode;
b. Delete the bottom face of of the cube;
c. Select the four vertices of the bottom of the cube;
d. Press 'E' for extrude and Esc to stop the extrude, then immediately press 'S' for Scale. As you move the cursor you should see the newly extruded edges move in and out in a square on the plane of the bottom of the cube;
e. Almost there!
f. Now apply a Solidify Modifier. The thickness should be whatever you wish it to be depending on the material you are hoping to print in. Leave the Offset at -1.